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A356904 E.g.f. satisfies A(x)^2 * log(A(x)) = x * (exp(x) - 1).

%I A356904 #24 Feb 16 2025 08:34:03
%S A356904 1,0,2,3,-32,-175,2376,29617,-371440,-9251919,91421560,4529155961,
%T A356904 -26677647864,-3160004989271,1541460644192,2946529440977865,
%U A356904 19556193589426336,-3498019439220155551,-56274505323609293208,5077223330715030358009
%N A356904 E.g.f. satisfies A(x)^2 * log(A(x)) = x * (exp(x) - 1).
%H A356904 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356904 a(n) = n! * Sum_{k=0..floor(n/2)} (-2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
%F A356904 E.g.f.: A(x) = Sum_{k>=0} (-2*k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
%F A356904 E.g.f.: A(x) = exp( LambertW(2 * x * (exp(x) - 1))/2 ).
%F A356904 E.g.f.: A(x) = ( -2 * x * (1 - exp(x))/LambertW(-2 * x * (1 - exp(x))) )^(1/2).
%t A356904 nmax = 19; A[_] = 1;
%t A356904 Do[A[x_] = Exp[((Exp[x]-1)*x)/A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
%t A356904 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o A356904 (PARI) a(n) = n!*sum(k=0, n\2, (-2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
%o A356904 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-2*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
%o A356904 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*x*(exp(x)-1))/2)))
%o A356904 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*x*(1-exp(x))/lambertw(-2*x*(1-exp(x))))^(1/2)))
%Y A356904 Cf. A355843, A356797, A356798, A356902.
%Y A356904 Cf. A349654.
%K A356904 sign
%O A356904 0,3
%A A356904 _Seiichi Manyama_, Sep 03 2022